Structural Induction on WFF For a formula α ∈ WFF we let `(α) denote the number of symbols in α that are left brackets ‘(’, let v(α) the number of variable symbols, and c(α) the number of symbols that are the corner symbol ‘¬’. For example in ((p1 → p2) ∧ ((¬p1) → p2)) we have `(α) = 4, v(α) = 4 and c(α) = 1. Prove by induction that he following property holds for all well formed formulas: • `(α) = v(α) + c(α) − 1

1. For 25 randomly selected students, (

a) how many ways do all the 25 students have different birthday?

b) find the probability that at least two students have the same birthday.

c) find the probability that only two share the same birthday.

d) for probability computed above identify the sample space, the experiment and outcomes.

e) can a random variable be used for the case above? If yes list the possible values for the variable.

I figured out a through c but i need D and E answered please!!

Show that if (1) F₁ and F₂ are connected sets, and (2) F₁ ∩ F₂ is not empty, then  F₁ ∪ F₂ is connected.

also

Suppose that F is connected. Show that F¯ (the closure of F) is also connected.

Show that id D is dense metric space of X and if all cauchy sequences Yn from a sense set D converge in D then X is complete.

Find the eigenvalues

λ_n

and eigenfunctions

y_n(x)

for the given boundary-value problem. (Give your answers in terms of n, making sure that each value of n corresponds to a unique eigenvalue.)

x²y'' + xy' + λy = 0,  y(1) = 0,  y'(e) = 0

Which of the following is not an assumption of the one-way ANOVA?

A. Group means are equal

B. Group variances are equal

C. data between and within groups are independent

D. Observations are normally distributed within groups.

1. police car is traveling south towards the intersection that a robber’s car has just crossed traveling east. The robber’s car is currently traveling 55 mph. The police car’s speed is currently 63 mph. How is the distance between the cars changing when the cop is 1.2 miles from the intersection, and the robber is .5 miles from it? Be sure to answer in a complete sentence.

2. In calm water, an oil spill spreads out in a circular fashion. Suppose the radius, r, of the spill is increasing by 3 feet per minute. How fast is the area, A, of the spill growing when the radius is 10 feet? Be sure to answer in a complete sentence.

3. We want to construct a box whose base is length is 5 times the width.  The material used to build the top and bottom cost $9 per sq. ft. and the material used to build the sides cost $7 per sq. ft.  If the box must have a volume of 75 cu. ft., determine the dimensions that will minimize the cost to build the box.

Write a program that asks the user for a file name.

The file contains a series of scores(integers), each written on a separate

line.

The program should read the contents of the file into an array and then

display

the following content:

1) The scores in rows of 10 scores and in sorted in descending order.

2) The lowest score in the array

3) The highest score in the array

4) The total number of scores in the array

5) The average score in the array.

6) The median score in the array.

Please find a file named scores.txt in Canvas.

Program structure shall be as follows:

A top level function named TestScoreAnalysis() which accepts a

file of scores to be analyzed and displayed to the console

screen describing the six content items specified above.

Each task shall be done in a separate function.

There are nine(9) tasks,... therefore at least nine functions

should be constructed to meet the requirements.

Allocate an array to hold at least 100 scores.

Note - The programming language is C++

subject:engineering mathematics

solve [cos(x+y)-sin(x+y)]dx-sin(x+y)dy=0,is the ODE exact?Find an integrating factor and solve the ODE.

A 25-year-old male presents to clinic complaining about eye pain and itching in his eyes. Upon examination the ophthalmologist found redness and some discharge of yellow pus. He looked into the retina to see if there were any abnormalities. He then cultured the pus and found a bacterial infection. He prescribed a fortified tobramycin ophthalmic solution of 0.5% w/v. The patient took the prescription to the pharmacist. The pharmacist looked at his shelves and found a tobramycin solution containing only 3 mg/mL concentration and a tobramycin injection containing 40 mg/mL. How many milliliters of this injection the pharmacist must add aseptically to a 5-mL container of the ophthalmic solution to prepare one 0.5% w/v in concentration

Using definitions of dot product and cross product, show that dot product and cross product are distributive (a) If the three vectors are coplanar. (b) in general.

also,

Is the cross product of two vectors associative i.e. A × (B × C ) = (A × B ) × C ? If so prove it. If not provide a counter example.

Let G be a cyclic group generated by an element a.

a) Prove that if aⁿ = e for some n ∈ Z, then G is finite.

b) Prove that if G is an infinite cyclic group then it contains no nontrivial finite subgroups. (Hint: use part (a))

Problem 3. Let F ⊆ E be a field extension.

(i) Suppose α ∈ E is algebraic of odd degree over F. Prove that F(α) = F(α^2 ). Hints: look at the tower of extensions F ⊆ F(α^2 ) ⊆ F(α) and their degrees.

(ii) Let S be a (possibly infinite) subset of E. Assume that every element of S is algebraic over F. Prove that F(S) = F[S]

diff eq

What, if anything, do the theorems of this chapter allow you to conclude about the existence and uniqueness of solutions to the following initial value problems.

y′′+ty′−t^2y=0; y(0)=0,y′(0)=0